The idea of using Riedtmann's well-behaved functors to study compositions of irreducible morphisms has been explored in a number of articles. Here we introduce the concept of mesh-comparable components of the Auslander-Reiten quiver, which are components for which a Riedtmann functor exists without the necessity of taking a covering, such as the universal or the generic one. We show properties of this type of component, and study the problem of compositions of irreducible morphisms in this context.